The framework in [1]} and [2]} gives that if \(W(t)\)  is
\(\tau \) -splittable for \(\tau = (\delta /2^t)^{O(1)}\) , then the above
instance of \(\textsf {MAX t-XOR} \)  can be solved to additive error \(O(\delta )\) 
using the Sum-of-Squares (SOS) SDP hierarchy.
Broadly speaking, splittability allows one to (recursively) treat
instances as expanding instances of problems with two “tuple
variables” in each constraint, which can then be analyzed using known
algorithms for 2-CSPs [3]}, [4]} in the SOS
hierarchy.
Combined with parity sampling, this yields a unique decoding
algorithm.
Crucially, this framework can also be extended to perform list
decodingWhile unique decoding can be thought of as
recovering a single solution to a constraint satisfaction problem, the
goal in the list decoding setting can be thought of as obtaining a
“sufficiently rich” set of solutions which forms a good cover. This is
achieved in the framework by adding an entropic term to the
semidefinite program, which ensures that the SDP solution satisfies
such a covering property. up to a radius of \(1/2 - \sqrt{\varepsilon }- \delta \)  under a similar condition on \(\tau \) , which will be very
useful for our application.
